Well, the only method I could see working is that if that definition worked, then it's negation must always be false, that: given , there exists such that and must always be false. If I can find a case where the negation is true, then obviously the negation is not always false and therefore the new definition cannot imply continuity. But that is as far as I have gotten so far.

Or I guess the new definition just shows that if , then but then I'd just have to show that that is not equivalent to if , then but I do not know how to go about that exactly.

Okay, so I showed that the new definition does not imply the sequential definition of continuity by using the example of when and on the domain . Kinda stuck on showing if the converse is also not necessarily true.